Like poles repel and opposites attract. Such is an elementary rule of thumb and one of the basic properties of magnetism: a magnet always has two 'inseparable' poles, north and south. Yet there is no fundamental reason why this should be the case, why does a magnet always have 2 poles? Why can't the field lines of the magnetic field have a terminating end? Why can't a magnet have only one pole? Since electric field lines terminate as electric charges, it seems as though there are simply no magnetic charges. Case closed? Not quite. In classical electrodynamics, Maxwell's equations have an elegant symmetry; the electric-magnetic duality (ensuring the electric and magnetic fields behave identically). This symmetry appears broken because no magnetic charges have been found but the existence of monopoles would solve this anomaly and restore the elegant symmetry. In quantum mechanics, the forces of electromagnetism are quantified in terms of scalar and vector potentials as opposed to electric and magnetic fields and their introduction seems to break the duality. Because electromagnetism has an abelian U(1) symmetry, one can perform a gauge transformation using an unlimited number of potentials to give rise to the same fields; however the vector potential seems to prohibit magnetic charges due to the disappearance of the divergence of the curl of a vector field. Dirac devised a means to apply a vector potential to construct a monopole; a means similar to Faraday who used a long magnet contained in a mercury-filled vessel in such a manner that one of the poles was beneath the surface while the pole above acted as a monopole. The existence of monopoles can explain the quantisation of electric charge and hence Dirac envisaged a semi-infinite solenoid with an end that possessed a non-zero divergence (thus acted as the monopole) and Dirac strings (infinitely thin flux tubes that connect two monopoles). Moving onto GUTs, the Weinberg-Salam unification incorporates a U(1) x SU(2) symmetry which is broken by the Higgs field at low energies; a simpler equivalent is the Georgi-Glashow SO(3) model. 't Hooft and Polyakov found that a solution to such a model exists that incorporates both electric and magnetic charges; their topologically stable solution involves a Higgs field of stationary length with varying direction in each different direction. And for the field to be continuous, a point-like flaw in the origin of the field can't be a vacuum state; thus the origin of the field is a clump of energy corresponding to a massive particle (since the Higgs field disappears at the origin, the SO(3) symmetry is left unbroken. Interestingly, such a particle possesses magnetic charge because electromagnetism is made by oscillations around the Higgs field vector, one can quantify the magnetic field and the 't Hooft-Polyakov solution turns out to be a monopole. Even though monopoles haven't been directly observed, let alone discovered, they play an important role in modern physics; especially in explaining the phenomenon of quark confinement in QCD. At extremely low temperatures, materials become superconductors and allow current to flow without resistance but eject any magnetic flux (Meissner effect); if we could put a monopole-anti monopole pair into a superconductor, what would happen? Since any magnetic flux is ejected, a way to resolve this dilemma would be that an Abrikosov-Gorkov flux tube forms between the pair, hence the flux is restricted to this tube. And since the flux tube has a nonzero energy value, the quantity of energy needed to separate the pair increases in a linear fashion. Finally, monopoles are important is cosmology because GUTs predict they were produced in the early universe; the Kibble mechanism is likely candidate for how that happened. It proposes the universe contains domain walls with arbitrary yet uniform field direction and the Higgs field inserts itself continually between a pair of domains but the field disappears in the origin causing topological defects. But when two pairs of domain walls meet, a monopole can be made.
No comments:
Post a Comment